Anti-persistent adherence dynamic of the COVID-19 vaccines

This research explores the multifractal dynamics of time series of the daily number of vaccinees for COVID-19, considering six European countries (Belgium, Denmark, France, Germany, Greece and Italy) using the Multifractal Detrended Fluctuations Analysis (MF-DFA). We calculate the multifractal spectrum f(α) and apply a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. We found that the multifractal dynamics of all these countries are characterized by strongly anti-persistent behavior (α 0 < 0.5) a lower degree of multifractality, and small fluctuations are dominant in the multifractal spectrum. From an immunization perspective, it means that a panorama that encompasses the population’s behaviour is marked by the dynamics of anti-persistent adherence to COVID-19 vaccines. Our findings confirm that the period of immunization of the population that adhered to the vaccination campaigns is short and that the application of new doses of vaccines must obey this phenomenology to keep people safe. In addition, we used the multifractal efficiency coefficient to rank countries that are most proactive in developing campaigns that promote greater adherence and loyalty to COVID-19 vaccines. Our findings indicate that Germany, Belgium and France were more efficient than Greece, Denmark and Italy.


Introduction
Developing a vaccine involves high costs with research and development (R&D), skilled human capital, time, and high failure rates [1]. In this way, the immunology paradigm of vaccine development is closely associated with several pauses to evaluate the data and many checks of conformity of the production process.
Specifically, the use of state-of-the-art genomic sequencing [13], reverse genetics and investment by the public, private and non-governmental sectors in biotechnology companies and the vaccine industry have collaborated to reduce vaccine development time, especially during epidemic emergencies [14].
This information must necessarily be in the public domain, which would tend to reduce people's hesitation still tied to the classic paradigm of vaccine development. In addition, it effectively contributes to the fight against the online anti-vaccine movement [15] and the COVID-19 vaccine hesitancy [16]. Note that both phenomena are usually linked to social media, which leads to a painful scenario of lack of culture, ignorance and obscurity.
Moreover, the permissibility of social media companies [15] to store this type of content impacts the global public health landscape and the massive spread of fake news. Lipic et al [17] exist a mathematical relation between the country's corruption and the number of COVID deaths alongside the fraction of anti-vaxxers. Thus, the higher the country's corruption displays, the higher the number of COVID deaths and the more people oppose the vaccination. This research simultaneously encompasses two topics highly relevant to Science and of interest to the general public. Specifically, we address the multifractal dynamics of COVID-19 vaccines by considering six European countries (Belgium, Denmark, France, Germany, and Greece) and the public adherence rate to COVID-19 vaccines.
Therefore, we apply the Multifractal Detrended Fluctuations Analysis (MF-DFA) method to consider the time series of the daily number of vaccinated. For each country, we investigate the Generalized Hurst exponent h (q) and the Rényi exponent τ(q) and quantify their statistical properties, which allowed us to examine separately the small contributing scale (primarily via the negative moments q) and the large scale (via the positive moments q). Moreover, we calculate the multifractal spectrum f (α) and employ a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process.
Our findings suggest that all countries show strongly anti-persistent behavior α 0 < 0.5. From an immunology point of view, this phenomenology is characterized by short memory. Thus, it sheds light that the period of immunization of the population that adhered to the vaccination campaigns presents short memory and that the application of new doses of vaccines must comply with this phenomenology to keep the people safe. The highlights of this paper are: (i) It promotes the synergy among Biophysics and Immunology; (ii) It present relevant insights into the multifractal dynamics of the COVID-19 vaccines for six European countries; (iii) It shows that these countries are characterized by a strongly anti-persistent behaviour α 0 < 0.5; (iv) It suggests that an inverse mathematical relationship between the number of vaccinated and COVID-19 mortality; (v) It reveals the multifractal efficiency for each country.
The remainder of this paper is organized as follows. Section 2 displays the data and the multifractal methods that we have applied in this research. Section ?, presents and discusses our results. Sections 3 and 4, formalizes our concluding remarks.

Data
We have investigated the daily number of vaccinated for COVID-19, considering six European countries (Belgium, Denmark, France, Germany, Greece, and Italy). For each country, the periods cover more than 19 months, from December 27th, 2020 until August 01, 2021 with 501 observations. The data were collected at https://ourworldindata.org/covid-vaccinations?country=OWID_WRL.
In the initial phase, referring to the investigation of the collected data, we apply descriptive analysis. Table 1 provides a list of details and descriptive statistics for each country.
Also, we employ the Box plot technique to examine the variation of observed data of a numerical variable through quartiles. Figure 1 shows the Box plot.
We apply a mathematical transformation called the first-order differentiation process. The most usual differentiation process consists of making successive differences from the original series to eliminate seasonality and trend so that it can become a stationary series. The first-order differentiation process is formalised by is the number of vaccinated for country i at time t.
The MF-DFA method follows a sequence of five steps [18]. Considering that d k is a time series of length N. Then, where i = 1,...,N andd is the mean value of the time series.
non-overlapping segments of equal length l.
(iii) For each N l segments, its estimates the local trend by fitting the least-squares of the series. So, the variance without trend is calculated by (iv) The average value for all segments is used to obtain the fluctuation function to the q by: where q can assume any real value, but different to 0 ( ) ¹ q 0 .
(v) Determine the scaling behavior of fluctuation functions by investigating log-log plots F q (l) versus l for each value of q. In the case of long-term correlations are present, F q (l) will increase with l as a power law where h(q) is obtained as the slope of the linear regression of log F q (l) versus log l.
The slope of the log F q (l) versus log l can suggest a family of scaling exponents h(q). Considering the values of h(q) can distinguish whether a process is fractal or multifractal [29]. Therefore, for monofractal time series h(q) is independent of q. While, for multifractal time series h(q) is a decreasing function of q. The Generalized Hurst exponent h(q) are related with the Rényi exponent by where τ(q) is the the Rényi expoent, q can assume positive and negative values. In the case of monofractal time series τ(q) is a linear function of q. While, for multifractal time series τ(q) is a non-linear function of q.
Another measure to investigate the multifractal properties considering a time series is estimated the multifractal spectrum f (α). The multifractal spectrum is given by The multifractal spectrum provides a description of the multifractal measure in terms of interlaced sets with singularity force α where f (α) is the dimension of the contour subset characterized by α. Taking into account a monofractal structure, the uniqueness of the spectrum produces a single point, while for multifractal structures,  the uniqueness of the spectrum is given by a downward concave function, whose degree of multifractality is evaluated by f (α).
Then, we fit the multifractal spectrum to a fourth-degree polynomial [30,31] to evaluate the complexity of the series. The fourth-degree polynomial is given by a a a a a a a a = + - To differentiate the multifractal spectrum f (α) quantitatively, it is also convenient to compute the width of the spectrum ( ) a a -W max min evaluated from equating the fitted curve to zero, and the skew parameter  can be used to measure of complexity where a series with a high value of α 0 , a wide range W of scaling exponents, and a right-skewed shape can be considered more complex than one with the opposite characteristics [32].

Empirical results
Although the countries analyzed are on the European continent, the differences between these countries' demographic, social, economic and cultural profiles are notorious. Figure 2 presents the time series of the temporal evolution of the daily number of vaccinated for COVID-19 and the first-order-differentiation process. Figure 5. The plots of the Rényi exponents for τ(q) (derived using τ(q) = qh(q) − 1) for these countries. The red dots reflect the firstorder-differentiation process time series. While the black dots reveal the shuffled series.
We use the MF-DFA method to explore the multifractal properties of the first-order-differentiation process time series. Figure 3 displays the fluctuation values exhibited by these time series.
We employ the shuffling procedure performed 1000 × N transpositions on each series and was repeated 1000 times with different random number generator seeds. Thus, we compute the Generalized Hurst exponent denoted by ( ) h q . Figure 4 depicted the Generalized Hurst exponents for the first-order-differentiation process time series and shuffled series considering these countries.
For all countries, our results reveal that the ( ) H q decreases a function of q. So, negative q values enhance small fluctuations, while large positive q values correspond to large fluctuations.  Figure 5 shows the Rényi exponents for the first-order-differentiation process time series and shuffled series considering these countries.
Moreover, we perform a fourth-order polynomial regression on the multifractal spectrum f (α) to specify the position of maximum α 0 and the zeros of the polynomial, a max and a min , which are employed to estimate the width of spectrum W and the asymmetry parameter r.
The parameter α 0 reflects the position of the maximum of f (α), and promotes a strong estimate of the overall Hurst exponent in which, a value of α 0 > 0.5 presents a persistent process, α 0 < 0.5 reflects an anti-persistent process and α 0 = 0.5 reveals a totally random process [24,30,31,33,34]. Figure 6 shows the multifractal spectrum for the first-order-differentiation process time series and shuffled series considering these countries.
Also, the measure of complexity parameters (α 0 , W and R) are present in table 2.
Based on the graphical analysis of figure 6 and the values of table 2, we observe that these countries are marked by strongly anti-persistent behaviour or anti-persistent long term correlations (α 0 < 0.5) a lower degree of multifractality, and small fluctuations are dominant in the multifractal spectrum.
The values of W indicates that Germany (W = 0.33879), Belgium (W = 0.497738), and Denmark (W = 0.550525) are less complexity than France (W = 0.635908), Italy (W = 0.651284) and Greece (W = 0.760815). The countries that exhibit the values of asymmetry parameter (R < 1) indicating a left-skewed spectrum and reflects that the multifractality is more influenced by large fluctuations and long-range correlations. Otherwise, the countries that display the values of asymmetry parameter (R > 1) indicating multifractality is more influenced by the scaling of small fluctuations and right-skewed spectrum.
The multifractal efficiency coefficient is measured through the absolute difference between the observed α 0 and the random (α 0 = 0.5). Thus, the higher the value of this index, the greater the proactivity of the country in developing campaigns that provided greater adherence and loyalty to the COVID-19 vaccines. The values of the efficiency coefficient reveal that Germany, Belgium and France were more efficient than Greece, Denmark and Italy.

Conclusions
In summary, we examine the multifractal dynamics for the daily number of vaccinated time series considering six European countries (Belgium, Denmark, France, Germany, and Greece) using the Multifractal Detrended Fluctuations Analysis (MF-DFA) method. We discover that all countries are display strongly anti-persistent behaviour α 0 < 0.5. Based on the immunology perspective, it suggests that this phenomenology is characterized by short memory (anti-persistent dynamic adherence to immunization). Thus, our findings categorically warn that the period of immunization of the population that adhered to the vaccination campaigns is short and that the application of new doses of vaccines must comply with this phenomenology to keep the people safe.
We emphasize that for Physics, a phenomenon marked by long-term memory is one in which an immediate occurrence reverberates for decades. It is widely known that there was a real need to renew vaccine doses, which must maintain mainly due to the mutating capacity of SARS-CoV-2 (new variants).
In this sense, our results suggest an inverse mathematical relationship between the number of vaccinated and COVID-19 mortality. Thus, when public adherence to COVID-19 vaccines increases, there is a decrease in the number of infected and lethality. However, the persistence of this decline in both series takes people back to their comfort zone related to a sense of security. It has begun to fail to comply with the recommendations of the World Health Organization (WHO). Consequently, infections and lethality increase again and this cycle repeats, which explains the analyzed time series's short-term memory and anti-persistent behaviour.
For each country, we present the efficiency multifractal coefficient that quantifies the proactivity of the country in developing campaigns that provided greater adherence and loyalty to the COVID-19 vaccines. Based on the values of this measure, we rank the countries. It indicates that Germany, Belgium and France were more Table 2. Multifractal parameters α 0 , W and r for the first-order-differentiation process time series and shuffled series considering these countries.

Ranking
Country efficient than Greece, Denmark and Italy. We suggest that our empirical evidence be used to combat fake news and resolve doubts associated with the effectiveness of COVID-19 vaccines.

Conflict of interest
The authors declare that this work has no conflicting personal or financial influences.